Sum-dominant sets and restricted-sum-dominant sets in finite abelian groups

David B. Penman ¹   ; Matthew D. Wells ¹

¹ Department of Mathematical Sciences University of Essex Wivenhoe Park Colchester CO4 3SQ, United Kingdom

Acta Arithmetica, Tome 165 (2014) no. 4, pp. 361-383

Résumé

We call a subset $A$ of an abelian group $G$ sum-dominant when $\def\abs#1{\vert#1\vert}\abs{A+A}>\abs{A-A}$. If $\def\abs#1{\vert#1\vert}\abs{A\mathbin{\hat{+}}A}>\abs{A-A}$, where $A\mathbin{\hat{+}}A$ comprises the sums of distinct elements of $A$, we say $A$ is restricted-sum-dominant. In this paper we classify the finite abelian groups according to whether or not they contain sum-dominant sets (respectively restricted-sum-dominant sets). We also consider how much larger the sumset can be than the difference set in this context. Finally, generalising work of Zhao, we provide asymptotic estimates of the number of restricted-sum-dominant sets in finite abelian groups under mild conditions.

Détail

DOI : 10.4064/aa165-4-6

Keywords: call subset abelian group sum dominant def abs vert vert abs abs a a def abs vert vert abs mathbin hat abs a a where mathbin hat comprises sums distinct elements say restricted sum dominant paper classify finite abelian groups according whether contain sum dominant sets respectively restricted sum dominant sets consider much larger sumset difference set context finally generalising work zhao provide asymptotic estimates number restricted sum dominant sets finite abelian groups under mild conditions